Note on desuspending the Adams map

Transcription

1 Math. Proc. Camb. Phil. Soc. (1986), 99, Printed in Qreat Britain Note on desuspending the Adams map BY F. R. COHEN AND J. A. NEISENDORFER University of Kentucky and Ohio State University (Received 9 May 1984; revised 17 May 1985) Let p denote a fixed prime and let P n (p r ) denote the cofibre of the degree p r map on consider n l).p 3 (p r ) and show that the map q in the cofibre sequence induces a split epimorphism on the ^-primary component of 7T 2p S 3 if p > 2. That analogous maps q: P n (p r )->-S n, n ^ 4, induce split epimorphisms on the jj-primary component of n n+2p _ 3 S n, p > 2, is shown in work of J. F. Adams [1]. It is the purpose of this note to document the above computation in the case n = 3 for the use of others. Recall that the generator a x of the ^-primary component of n ip S 3,p > 2, has orders and thus extends to a map P 2p+1 (p)->s 3. The result of this note is that, if p is an odd prime, there is a choice of map A: P 2p+1 (p) -> P 3 (p r ) such that qoa induces an epimorphism on the ^-primary component of n 2p S 3. In [1], Adams constructed a map with n greater than 3 such that qoa induces an epimorphism on the ^-primary components of Jr n+2p _ 3 S n. Thus the lift constructed here is a choice of desuspension of the Adams map. We give two proofs, one of which is elementary but longer. 1. The short proof Throughout this section, p is an odd prime. We wish to show PROPOSITION 1-1. There is a homotopy commutative diagram A S 2p where a^ is a generator for the p-primary component ofn 2p S 3. The proof of 1-1 occupies the rest of Section 1. Let S 2n+1 {p r } be the homotopy-theoretic fibre of the degree p r map on S Zn+1. Note that the pinch map q: P 2^1^) -> S 2n+1 factors thiough a map q': since p r q is null homotopic. Recall ([2], [6]) that there is a fibration sequence 00 fi«' G(n) x n * fc=l

3 Note on desuspending the Adams map 61 The main result of this note is given by the second statement in PROPOSITION 2-1. Letp be an odd prime. (a) There exists a map p: P 2p+1 (p) -> X such that the composite jkp induces an isomorphism on H 2p ( ; Z/p). (b) There exists a homotopy commutative diagram where a x is a generator for the p-primary component of n 2p 8 3. Next consider M(p r ) = (V 0 *ij<zp-2 pi+2i (P r )) V P 2p {p T+1 ). We shall show LEMMA 2-2. Ifp > 2, there exists a map A: M(p r )-> X such that (i) A induces an integral homology isomorphism in degree less than 2p and (ii) A induces a split monomorphism on n i for i less than or equal to 2p. We need to know the integral cohomology of X in a range. LEMMA 2-3. Ifp > 2, then (Z/p r if i = 2k, 4 s? 2k < 2^-2, if i = 2p + 1 or 2p + 2, and,0 for other i, 0 < i < 2p + 2. Furthermore, (jk)* is an isomorphism on H 2p+1 ( ; Z). Granting 2-2 and 2-3, we prove 21 (a). Let K denote the cofibre of A, the map which A 0 was given in 2-2. Consider the cofibre sequence M(p r ) -> X -* K. Let F denote the homotopy-theoretic fibre of 6. The natural lift of A to F induces an integral homology isomorphism in dimensions < 2p + 2 by the Serre exact sequence. Thus by the long exact homotopy sequence for a fibration together with 2-2, there are split short exact sequences 0->7 for i < 2p. But H 2p (K; Z) is isomorphic to n 2p K by the Hurewicz theorem. Thus n 2p K is isomorphic to Z/p. Any choice of splitting for the epimorphism n 2p X -> n 2p K induces a map p: S 2p -> X which is of order p. Thus there is a choice of extension of p to p. pnp+i(p) _>. x. By 2-3, jkp induces an isomorphism on H 2p ( ; Z) and 2-1 (a) follows. To prove 2-1 (b), observe that a, x : S 2p -» $ 3 <3> has non-zero Hurewicz image. This suffices. Observe that the above proof gives PROPOSITION 2-4. Ifp > 2, then M(p r ) if if i < 2p. We omit the proof of the following 2-primary analogue: i = 2p, and

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